          t-angles.sty ^{∗}

(Diagrammatic macros for tangles and braided Hopf algebras)

Yu. Bespalov  V. Lyubashenko

Version 04.04.99

Contents

1 Introduction 1 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Main features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Macros in pictures 3 Straight lines and nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 (Under/over)crossings. Braiding and symmetry . . . . . . . . . . . . . . . . . . . . . . 3 (Co)pairings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Morphisms. Frame and dash boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 (Co)multiplications and cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 (Co)actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Examples 5

4 Development 5 History and versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Directions for modification ^{1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

A Exercises 6

1 Introduction

Usage:

   \usepackage[emtex]{t-angles}

(for emtex drivers, dviwin and for new dvips ) or

                                           \usepackage{t-angles} ≡
                                           \usepackage[TPIC]{t-angles}

(for TPIC drivers such as

                                          dviwin,  xdvi,  dvips )

∗t-angles.sty is available from http://www.math.ksu.edu/∼lub/ 1What can you suggest?

To use with kluwer.cls add the option kluwer :

\usepackage[emtex,kluwer]{t-angles} or \usepackage[kluwer]{t-angles}

Main features:

• The environments

             {tangle} {tanglec} {tangler} {tangles}

are arrays with one or more column style:

                                          {array}[l] {array}[c] {array}[r] {array}

(left) (centered) (right) (any)

respectively.

• \unitlens is the global length parameter. Default value is 10 pt .

• \hstretch and \vstretch are relative length parameters, horizontal and vertical stretch:

                    \unith = \hstretch % of \unitlens ,
                    \unitv = \vstretch % of \unitlens ,

set to an integer number of percents before the beginning of a tangle.

Default settings are \hstretch 100 and \vstretch 100. The commands \hstretch

and \vstretch can be used only outside of tangle environments (with an exception of embedded tangle environments).

• The commands \hstr{<number>} , \vstr{<number>} can be used inside tangle

environments instead of \hstretch<number> , \vstretch<number> . They will act

within their L^{A}TEX scope.

• The height of every row is 2 \unitv or \unitv if the command \hh (see below) is used; the widths of standard fragments are 0, .5, 1, 2, 3 or 4 \unith

• The command \hh obeys to L^{A}TEX scope rules. The command \HH acts in the same way as \hh but put at the beginning of a row works for the whole row in the {tangles} environment.

• The style understands the commands \thinlines and \thicklines .

• The command \step[<number>] is used to produce horizontal space \kern <number> \unith and works in any mode (inside and outside of the tangle environment).

                \step = \step[1]       \Step = \step[2]
                \hstep = \step[.5]     \hhstep = \step[-.5]

• Vertical spacing before the next row is produced by \\[⟨vertical space⟩] with optional argument (like in standard {array} environment).

• The command \object#1 is used to put the object #1 directly over or under the end of the string (inside and outside of the {tangle} environment). It adds a vertical space below or above as required. More space can be added as above.

The command \Put(x coord,y coord)[binding point]<object> puts <object> into

the intended position and works in {tangle[cs]} environment like a combination of \put and \makebox . Coordinates are integers, measured in .1 \unith , .1 \unitv units; binding point is a combination of two letters lcr and tcb according to the usual L^{A}TEX rules.

The commands \nodeu#1,   \noded#1,   \nodel#1,   \noder#1,   \noderu#1,
\noderd#1,   \nodelu#1,   \nodeld#1 produce zero boxes and put #1 into the

corresponding position. The picture

.. . . .. ..

is described by the lines:

                                \vstretch 150  \hstretch 150
                                \begin{tangle}
                                \nodeu.\noded.\nodel.\noder.
                                \noderu.\noderd.\nodelu.\nodeld.
                                \end{tangle}

2 Macros in pictures

Straight lines and nodes

• The commands \id , \n , \s , \node , \unit , \counit work in hh-mode .

     [figure: \id; \idash; \n; \s; \nd; \sd; \node; \unit; \counit]

The command \FillCircDiam denotes the filled circle diameter. It is set to an integer between 1 and 9 (here the measure unit is 0.1 \unith ). Default value is 4.

• The command \hln <number> produces horizontal line on <number> \unith .

• Argument of \ne , \nw , \se , \sw , \ned , \nwd , \sed , \swd is 0,1,2,3 or

4; 0 produces empty box and other produce (dash) lines with horizontal projections equal

<argument> \unith cribbed into 1×2 box. The commands \ne , \nw , \se , \sw

produce 1×1 box in hh-mode .

    \nw1

            \nwd1

                    \sw0

                            \se3

                                    \sed3

                                            \nw3\nw2\nw1\n\ne1\ne2\ne3

• All the following commands work in hh-mode and produce the similar diagrams in .5×1 boxes.

                   \d    \dd    \hd    \hdd    \dh    \ddh

(Under/over)crossings. Braiding and symmetry

The commands \X , \XX , \x , \xx work in hh-mode and produce the similar diagrams of half width and height (1×1 boxes).

                 \x     \xx    \hx    \hxx    \X    \XX

(Co)pairings.

The commands \ev , \coev work in hh-mode and produce the similar diagrams of half width and height (.5×1 boxes). For convenience in hh-mode \hev ≡ \ev and \hev ≡ \ev .

              [figure: \hev; \ev; \Ev; \EV]

            [figure: \hcoev; \coev; \Coev; \COEV]

Morphisms. Frame and dash boxes

•

           [figure: f; \Q f; f; \QQ f; f; \O f; f; \morph f; f; \tu f; f; \td f; S; \S; S^{−1}; \SS]

•

                [figure: f; \ox f; p; \ro p; q; \coro q; p; \Ro p; q; \coRo q]

   • The commands \dbox#1#2,  \ffbox#1#2,  \obox#1#2,  \tbox#1#2 put $#2$ in the

middle of #1×2 (or #1×1 in hh-mode ) box with dash, rectangle, oval frame or without frame.

For example, the text

     \begin{tangles}{rcl} \HH\obox 10&&\obox 10\\ \HH\d&&\dd\\ &\hhstep\obox 3V\hhstep& \end{tangles}  produces  [figure: 0; 0; V]

(Co)multiplications and cocycles

The commands \cu , \cu*, \cd , \cd* work in hh-mode and produce the similar diagrams of half width and height (.5×1 boxes). For convenience in hh-mode \hcu ≡ \cu and \hcd ≡

\cd .

                     [figure: \Cu; \cu; \hcu; \hdcu; \hddcu]

                  [figure: \Cu*; \cu*; \hcu*; \hdcu*; \hddcu*]

                  [figure: \Cd*; \cd*; \hcd*; \hdcd*; \hddcd*]

                     [figure: \Cd; \cd; \hcd; \hdcd; \hddcd]

(Co)actions

Commands \lu , \ld , \ru , \rd have optional parameter [#1] which equals to width of the box:

   \hh   [figure: \Lu≡\lu[2]; \lu≡\lu[1]; \hlu; \hru; \ru≡\ru[1]; \Ru≡\ru[2]]

         [figure: \Lu≡\lu[2]; \lu≡\lu[1]; \hlu; \hru; \ru≡\ru[1]; \Ru≡\ru[2]]

         [figure: \Ld≡\ld[2]; \ld≡\ld[1]; \hld; \hrd; \rd≡\rd[1]; \Rd≡\rd[2]]

   \hh   [figure: \Ld≡\ld[2]; \ld≡\ld[1]; \hld; \hrd; \rd≡\rd[1]; \Rd≡\rd[2]]

Compositions

                      [figure: \kk; \luld; \cucd; \rurd; \k]

3 Examples

      [figure: B B; B B]= [figure] ; [figure]  is produced by: \begin{tangle} \object{B}\step\object{B}\\ \cucd\\ \object{B}\step\object{B} \end{tangle} \;=\enspace \begin{tangles}{lcr} \HH\cd&&\cd\\ \HH\id&\x&\id\\ \HH\cu&&\cu \end{tangles}

4 Development

History and versions

The style is produced by the first author in 1994 and used in [?, ?]. It was completely modified and essentially improved by the second author for applications in [?].

Directions for modification ^{2}

• In the future some problems can be solved by introducing global (logical) parameters that switch configuration and behavior of certain families of commands in questionable situations.

2What can you suggest?

• To adopt commands like in {picture} environment to produce special fragments of one time use.

• To make the second argument of the command \Put(#1)[#2]#3 optional.

• To produce command index for this manual.

• To add possibility to change size of circle in circled morphisms (in particular, to turn \morph into a special case of \O ).

Acknowledgments

Optional parameter for (co)actions is proposed by Bernhard Drabant. t-angles.sty contains parts of emlines2.sty by Georg Horn and Eberhard Mattes and parts of eepic.sty by Conrad Kwok.

Suggestions are welcomed.

References

A Exercises

How to produce the following ?

[figure: 0; 0; V]